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Boundedness of Lusin-area and functions on localized BMO spaces over doubling metric measure spaces. (English) Zbl 1220.42014
Let $(X,d)$ be a metric space endowed with a regular Borel measure $\mu$ such that $0<\mu(B(x,r))<\infty$ $(x\in X$, $r>0)$. $(X,d,\mu)$ is called a doubling metric measure space if $\mu(B(x,2r))\le C_1\mu(B(x,r))$. A positive function $\rho$ on $X$ is called admissible if, for some $C_0$, $k_0>0$, $\rho(x)^{-1}\le C_0\rho(y)^{-1}(1+\rho(y)^{-1}d(x,y))^{k_0}$ $(x,y\in X)$. For an admissible function $\rho$ and $q\in [1,\infty)$, a function $f\in L_{\text{loc}}^q(X)$ is said to be in $\text{BMO}_\rho^q(X)$ if $$\align\|f\|_{\text{BMO}_\rho^q(X)}= &\sup_{B=B(x,r);\,x\in X,\, r<\rho(x)}\biggl( \mu(B)^{-1}\int_B|f(y)-f_B|^q\,d\mu(y)\biggr)^{1/q}\\ +&\sup_{B=B(x,r);\,x\in X,\, r\ge\rho(x)}\biggl(\mu(B)^{-1}\int_B|f(y)|^q\,d\mu(y) \biggr)^{1/q}<\infty.\endalign$$ Similarly, a real valued function $f\in L_{\text{loc}}^q(X)$ is said to be in $\text{BLO}_\rho^q(X)$ if, in the definition of $\text{BMO}_\rho^q(X)$, $f_B$ is replaced by $\text{ess  inf}_{u\in B}f(u)$. In the case $(X,d,\mu)= (\Bbb R^n,|\cdot|,dx)$ and $\rho\equiv1$, $\text{BMO}_\rho^q(X)$ coincides with the Goldberg’s local BMO space bmo. If $\rho$ is an admissible function, $\text{BMO}_\rho^q(X)\cong\text{BMO}_\rho^1(X)=:\text{BMO}_\rho(X)$. Now let $\rho$ be an admissible function and $\{Q_t\}_{t\ge0}$ be a family of operators bounded on $L^2(X)$ with integral kernels $\{Q_t(x,y)\}_{t\ge0}$ satisfying that there exist $C$, $\delta_1>0$, $\delta_2\in (0,1)$, $\gamma>0$ such that $|Q_t(x,y)|\le C(\mu(B(x,t))+\mu(B(x,d(x,y)))^{-1}(1+d(x,y)/t)^{-\gamma} (1+t/\rho(x))^{-\delta_1}$, and $|\int_XQ_t(x,z)\,d\mu(z)|\le C(1+\rho(x)/t)^{-\delta_2}$. Using this family of operators, the authors define the Littlewood-Paley $g$-function $g(f)$, Lusin’s area function $S(f)$ and the $g_\lambda^*$ function $g_\lambda^*(f)$ by $$\align g(f)(x)&= \bigg(\int_0^\infty |Q_t(f)(x)|^2\,dt/t\bigg)^{1/2},\\ S(f)(x)&= \biggl(\int_0^\infty \int_{d(x,y)<t} |Q_t(f)(y)|^2 \mu(B(y,t))^{-1}\,d\mu(y)\,dt/t\biggr)^{1/2},\\ g_\lambda^*(f)(x)&= \biggl(\iint_{X\times(0,\infty)} |Q_t(f)(y)|^2 (1+d(x,y)/t)^{-\lambda}\mu(B(y,t))^{-1}\,d\mu(y)\,dt/t\biggr)^{1/2}.\endalign$$ Their main result is the following: Let $X$ be a doubling metric measure space satisfying one more condition, “the $\delta$-annular decay property”. Let $\rho$ be an admissible function. Assume that the Littlewood-Paley $g$-function is bounded on $L^2(X)$. Then $\|S(f)^2\|_{\text{BLO}_\rho(X)}\le C \|f\|_{\text{BMO}_\rho(X)}^2$. If $3n<\lambda <\infty$, the same result holds for the $g_\lambda^*$ function without assuming “the $\delta$-annular decay property”. Relating to their boundedness results, the authors give a nonnegative $f\in\text{bmo}(\Bbb R)$ which is not in $\text{blo}(\Bbb R)$.

42B25Maximal functions, Littlewood-Paley theory
42B30$H^p$-spaces (Fourier analysis)
51F99Metric geometry
Full Text: DOI arXiv
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